%I #29 Aug 21 2024 11:25:16

%S 4,108,337500,277945762500,79301169838123235887500,

%T 24018350267611933650627567399079537500,

%U 19868946365457062696924774946056904675112420776003728137500

%N a(n) = Product_{i=1..n} prime(i)^prime(i).

%C Denominator of Sum_{i=1..n} 1/(p(i)^p(i)), where p(i) = i-th prime. The numerators are in A117579. E.g., 1/4, 31/108, 96983/337500, 79870008269/277945762500, ... - _Jonathan Vos Post_, Mar 29 2006

%C Equally, denominator of Sum_{k=1..n}(-1)^(k+1) * 1/p(k)^p(k), where p(k) = prime(k). - _Alexander Adamchuk_, Aug 22 2006

%C C = Sum_{k>=1} (-1)^(k+1)/(prime(k)^prime(k)) = 1/2^2 - 1/3^3 + 1/5^5 - 1/7^7 + 1/11^11 - 1/13^13 + ... A122147 is the decimal expansion of C = 0.213281748700785698255627... - _Alexander Adamchuk_, Aug 22 2006

%C Hyperprimorials, from primorials by analogy with hyperfactorials. See A006939. - _Matthew Campbell_, Jul 30 2015

%F log a(n) ~ (n^2 log^2 n)/2. - _Charles R Greathouse IV_, Sep 14 2015

%e A122148(n)/a(n) begins 1/4, 23/108, 71983/337500, ... - _Alexander Adamchuk_, Aug 22 2006

%t Table[Denominator[Sum[1/Prime[k]^Prime[k],{k,1,n}]],{n,1,10}] (* _Alexander Adamchuk_, Aug 22 2006 *)

%t Denominator[Accumulate[1/#^#&/@Prime[Range[10]]]] (* _Harvey P. Dale_, Jan 24 2013 *)

%o (PARI) a(n)=prod(i=1,n,prime(i)^prime(i)) \\ _Charles R Greathouse IV_, Aug 05 2015

%Y Cf. A051674, A122147, A122148, A094289, A117579, A076265, A000040.

%K nonn,frac,easy

%O 1,1

%A _Jeff Burch_, Nov 23 2002

%E Entry revised by _N. J. A. Sloane_, Apr 10 2006

%E Edited by _N. J. A. Sloane_, Aug 04 2008 at the suggestion of _R. J. Mathar_